Hat die Währmeleitfähigkeit von Metall etwas mit dem Schmelzpunkt zu tun?

Jein.

Actually, the metallic thermal conductivity is dominated by the electron gas. Metals are metals because the atoms are bound in a certain manner in which the binding electrons are very freely movable. These can absorb and transmit thermal energy.

In addition, there is also the phononic thermal conductivity which passes through atomic nuclear vibrations in the crystal lattice. The atomic nuclei are “stressed” in the solid body by the bonds and can oscillate within this potential to absorb (thermal) energy. These vibrations excite adjacent atomic nuclei and thus heat can also be passed through a material.

If these so-called lattice oscillations become too strong, the bonds between the metal cores break up in the crystal and the metal becomes liquid. As a result, the atomic nuclei achieve new degrees of freedom and can conduct the heat as well as other liquids.